Understanding the Meaning of Discriminant

In the Derivation of the Quadratic Formula, we have learned that the solutions to the quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, is described by the equation

$x = \displaystyle\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

Graphically, getting the solutions of $ax^2 + bx + c = 0$ is equivalent to getting the value of $x$ when $y=0$ of the function $f(x) = ax^2 + bx + c$, $a \neq 0$. This means that the quadratic formula above describes the root of the quadratic function $f$.

From the equation, the $\pm$ indicates that the quadratic polynomial can have at most two roots, depending on the value of the expression under the radical sign. That is, the roots of the quadratic polyonomials are

$x = \displaystyle\frac{-b + \sqrt{b^2-4ac}}{2a}$

and

$x = \displaystyle\frac{-b - \sqrt{b^2-4ac}}{2a}$.

The number of roots, however, will depend on the expression $b^2 - 4ac$. Notice that if it is negative, there is no root (real root to be exact), since we cannot extract the square root of a negative number. Therefore, if $b^2 - 4ac < 0$, then, there is no real root. » Read more